[Inquiry] Re: Logic Of Relatives -- Commentary

[Jon Awbrey]

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LOR.  Commentary Note 4

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A wealth of issues arise here that I hope
to take up in depth at a later point, but
for the moment I shall be able to mention
only the barest sample of them in passing.

The two papers that precede this one in CP 3 are Peirce's papers of
March and September 1867 in the 'Proceedings of the American Academy
of Arts and Sciences', titled "On an Improvement in Boole's Calculus
of Logic" and "Upon the Logic of Mathematics", respectively.  Among
other things, these two papers provide us with further clues about
the motivating considerations that brought Peirce to introduce the
"number of a term" function, signified here by square brackets.
I have already quoted from the "Logic of Mathematics" paper in
a related connection.  Here are the links to those excerpts:

http://suo.ieee.org/ontology/msg04350.html
http://suo.ieee.org/ontology/msg04351.html

In setting up a correspondence between "letters" and "numbers",
my sense is that Peirce is "nocking an arrow", or constructing
some kind of structure-preserving map from a logical domain to
a numerical domain, and this interpretation is here reinforced
by the careful attention that he gives to the conditions under
which precisely which aspects of structure are preserved, plus
his telling recognition of the criterial fact that zeroes are
preserved by the mapping.  But here's the catch, the arrow is
from the qualitative domain to the quantitative domain, which
is just the opposite of what I tend to expect, since I think
of quantitative measures as preserving more information than
qualitative measures.  To curtail the story, it is possible
to sort this all out, but that is a story for another day.

Other than that, I just want to red flag the beginnings
of another one of those "failures to communicate" that
so dogged the disciplines in the 20th Century, namely,
the fact that Peirce seemed to have an inkling about
the problems that would be caused by using the plus
sign for inclusive disjunction, but, as it happens,
his advice was overridden by the usages in various
different communities, rendering the exchange of
information among engineering, mathematical, and
philosophical specialties a minefield in place
of mindfield to this very day.

Jon Awbrey

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